Sparse Approximation Via Iterative Thresholding

TL;DR: This paper studies two iterative algorithms that are minimising the cost functions of interest and adapts the algorithms and shows on one example that this adaptation can be used to achieve results that lie between those obtained with Matching Pursuit and those found with Orthogonal Matching pursuit, while retaining the computational complexity of the Matching pursuit algorithm.

TL;DR: The developed theory provides a successful practice of extension of the well- known Moreau's proximity forward-backward splitting theory to the L1/2 regularization case and verify the convergence of the iterative half thresholding algorithm and provide a series of experiments to assess performance.

TL;DR: This monograph covers proximal methods, block-coordinate descent, reweighted l2-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provides an extensive set of experiments to compare various algorithms from a computational point of view.

TL;DR: In this paper, the authors bring together machine learning, engineering mathematics, and mathematical physics to integrate modeling and control of dynamical systems with modern methods in data science, and highlight many of the recent advances in scientific computing that enable data-driven methods to be applied to a diverse range of complex systems, such as turbulence, the brain, climate, epidemiology, finance, robotics, and autonomy.

TL;DR: This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries and develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal.

Abstract: We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted l^p-penalties on the coefficients of such expansions, with 1 < or = p < or =2, still regularizes the problem. If p < 2, regularized solutions of such l^p-penalized problems will have sparser expansions, with respect to the basis under consideration. To compute the corresponding regularized solutions we propose an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. We also review some potential applications of this method.

TL;DR: This work develops a simple iterative procedure for the BPDN minimization that amounts to stepwise shrinkage, and demonstrates how the simple shrinkage emerges as the first iteration of this novel algorithm.

TL;DR: MCA is extended to a multichannel MCA (MMCA) for analyzing multispectral data and a range of examples which illustrates the results are presented.